Weak Convergence of sequences of functions.

Question

I'm trying to show that the following assumption found in Analysis, by Lieb and Loss, page 56, is true: f_k wanders off to infinity, an example, if f(x)=g(x+k) for some g in Lp space, implies that f_k converges weakly to zero, but doesn't converge strongly. I tried to compute the Lq norm of g, but I don't know how the existence of k in the argument of g effects this norm. Please help me.

Answer 1

Lebesgue measure is transaltion invariant so $\int \psi (x+k)dx=\int \psi (x)dx$. In aprticular $\|g(x+k)\|_p=\|g\|_p$ for each $k$. Suppose $f_k(x)=g(x+k)$ for some $g \in L^{p}$ with $1\leq p <\infty$. Let $h \in L^{q}$ where $\frac 1p +\frac 1 q =1$. Consider $\int f_k(x)h(x)dx=\int g(x+k)h(x)dx$. Given $\epsilon >0$ there exists a continuous function $\phi$ with compact support such that $\|g-\phi\|_p <\epsilon$. Note that $|\int g(x+k)h(x)dx-\int \phi (x+k)h(x)dx|\leq \epsilon \|h\|_q$ by Holders inequality. Hence it suffices to show that $\int \phi (x+k)h(x)dx \to 0$ as $ k \to \infty$. By DCT $\int_{|x| \leq M} \phi(x+k)h(x)dx \to 0$ for each $M$. By Holder's inequality $\int_{|x| >M} \phi (x+k)h(x)dx \leq \|\phi\|_p \|hI_{|x| >M}\|_q \to 0$ as $M \to \infty$. This completes the proof.